Use row operations to reduce the matrix to "upper triangular". That is, a matrix that has only 0's below the main diagonal. You do not need (nor want) to get 1 on the diagonal. If you reduce a matrix to upper triangular using only the row operations of "swap two rows" and "add (or subtract) a multiple of one row to (from) another", and not "multiply (or divide) one row by a number", then the determinant is just the product of the numbers on the main diagonal.